(J, J’bLossless Factorization for Descriptor Systems · The (J, ]‘)-lossless factorization for...

(J, J’bLossless Factorization for Descriptor Systems

Kin Kin* and Hidenori Kimura

Department of Mechanical Engineering for Computer-Controlled Machine y

Faculty of Engineering

2-1, Yamudaoka, Suita,

Osaka 565, Japan

Submitted by Yutaka Yamamoto

ABSTRACT

The (J, ]‘)-lossless factorization for descriptor systems based on the theory of conjugation is studied. The results on (J, I’)-1 oss ess 1 factorization for proper functions

are extended to descriptor systems, and those on J-lossless factorization for square

matrices are extended to nonsquare cases. Since (J, I’)-lossless factorization contains both the inner-outer factorization and the Wiener-Hopf spectral factorization as special cases, and the descriptor system contains the state-space system as a special case, the results obtained here unify the previous results on factorization problems.

Also, (J, J’)-1 oss ess factorization can be used to solve d-block H” control problems 1 with jw-axis zeros.

1. INTRODUCTION

The motivation of this paper comes from the study of the d-block H”

control problem with jo-axis (including infinity) zeros. It is now well known that the (J, J’>-1 oss ess 1 factorization plays a central role in H” control theory

(e.g. [ll, 121, and [201>. M oreover, it gives a simple and unified framework of

H” control theory from the viewpoint of classical network theory. Before discussing the (J, J’)-1 oss ess 1 factorization, we introduce the con-

cept of (J, J’)-1 oss ess matrices. Assume that integers m, r, p, and q satisfy 1

*Current address: Research Institute of Automation, Southeast University, Nanjing 210018, P. R. China.

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1290 XIN XIN AND HIDENORI KIMURA

m > p > 0 and r > q > 0. A matrix O(s) E RL~m+,~x~p+q~ is said to be (Jm,, JP,)-unitary if it satisfies

@ - (s)lmP(s) = Jp4 vs, (1)

where 0 - (s) := @r( -s), and

A (Jm,., JPq)-unitary matrix O(s) is said to be (Jmr, JPq)-ZossZess if it satisfies

where O*(s) := @r(s). If Jmr =JP4 (m = p, r = q), O(s) is said to be J,,-lossless. For the sake of notational simplicity, we write J = Jmr and

1’ =lpq in what follows. The interest in (J, J’>-1 oss ess 1 matrices is by no means new in the

literature 16, 11, 17, 181. An example of (J, J’)-lossless matrices is given by inner matrices. By an inner matrix, we mean a matrix G(s) E RH” that satisfies G _ (s)G(s) = 1. From the maximum-modulus theorem, we know that G*(s)G(s) < Z for s in Re[s] > 0. Therefore, an inner matrix is a special case of a (J, I’)-lossless matrix with r = q = 0.

The previous definition of (J, J’)-1 oss ess 1 factorizations is expressed as follows: a matrix G(s) is said to have a (1, Jr>-1 oss 1 ess factorization if it can be represented as a product G(s) = @(s)II(s), where O(s) is (J, J’)-lossless, and both II(s) and II-‘(s) are in RH”. The notion of (1, J’)-lossless factorization is rather general. Obviously, it is a generalization of the inner- outer factorization of stable matrices, which is well known in different application areas [13], to RL” matrices. It includes the Wiener-Hopf factorization for a positive matrix as a special case, and the relationship has been discussed by [18]. Th us, (J, I’)-lossless factorizations are of great importance in system theory. A necessary and sufficient condition for general unstable G(s) to have a (J, J’)-1 oss ess 1 factorization was derived in 1183, based on the theory of conjugation developed in [16].

However, the above definition of (J, I’>-1 oss ess 1 factorizations implies that they can only be applied to left invertible functions in RL”. Thus, the factorization cannot be applied to strictly proper functions or general rational functions. In order to treat the (J, I’>-1 oss ess 1 factorizations of these functions, we have to modify the previous definition of the (J, J’)-lossless

(J, J’hLOSSLESS FACTORIZATION 1291

factorizations. Instead of the requirement that both II(s) and n-‘(s) be in RH”, we only require that II(s) and K’(s) have full row ranks for every s in the open right half plane. Such a factorization will still be termed a

(J, J’N oss ess a I f ct orization in this paper. It is well known that the state-space representation can be used to

describe only (strictly) proper rational matrices, while the descriptor-form representation can be used to describe any rational matrix. In [I5], the notion of so-called extended ]-lossless outer factorizations for descriptor systems was studied, and a necessary and sufficient condition for the existence of an extended J-lossless outer factorization was expressed in terms of two generalized Riccati equations, which can be solved as generalized eigenvalue problems. However, the results are only for square rational matrices. They can only be applied to solve the l-block H” control problem with jw-axis zeros.

In [24], the notion of infinite zero compensation in [4] was used to study the (J, Jr>-1 oss ess 1 factorization for strictly proper functions. A simplified state-space realization of II(s) with the same dimension as G(s) was derived. It was found that the construction of the infinite zero compensator was not required. It was shown that the relations between eigenstructures of the system matrices and Hamiltonian matrix pencils of G(s) played an important role in the factorization.

The notion of (J, J '>-1 oss ess 1 factorizations for descriptor-form systems is discussed in this paper, based on the concept of J-lossless conjugation. The paper is organized as follows: we briefly review some previous results on the descriptor systems and J-lossless conjugation in Section 2. The eigenstructures of system matrices and Hamiltonian matrix pencils are studied in Section 3. In Section 4, (J, J’)-1 oss ess 1 factorizations for stable rational matrices are discussed. (J, J’>-1 oss ess 1 factorizations for general unstable rational matrices are discussed in Section 5.

NOTATION.

C(sZ-A)-‘B+D:= A ’ H--l C D'

C(sE - A))‘B + D := [w], E#Z,

C,, C_ = open right and left half plane, respectively, CI, R, = jo-axis and j~axis with infinity, respectively, R mXr = set of all m X r constant real matrices,

R(s) mX r = set of all m X r- rational matrices.


RL”,,, = set of all m X r rational proper matrices without pole on the jw-axis,

RH”,,, = set of all m x T rational stable proper matrices, deg G(s) = McMillan degree of G(s),

p(X) = maximum eigenvalue of X, Im{P} = {y E R” : y = Px, x E R”, P E Rmxn),

v{ -SE + A; D} = generalized eigenspace of -SE + A corresponding to the eigenvalues in domain D.

2. PRELIMINARIES

2.1. Matrix Pencils and Descriptor Systems We first briefly introduce the eigenstructure of matrix pencils from [21I

and [4]. It is known that there exist nonsingular matrices E and L which yield the Weierstrass form of the regular pencil -SE + A [det(sE - A) is not a constant] as follows:

-sI + A, 0 L( -SE + A)= = o 1 -sA,-kl ’ (3)

where hf is a Jordan matrix whose eigenvalues correspond to the zeros of the polynomial equation det( -SE + A) = 0, and A, is a nilpotent matrix in Jordan form. Decompose B = [8, &,,I in accordance with (3). Then the finite eigenstructure and infinite eigenstructure of -SE + A are defined by

AEf = E+Q, (4)

A&A, = E&, (5)

respectively. It has been shown that the eigenstructure can still be defined by (4) and

(5) when -SE + A is a pencil of full column rank [4]. Next, consider the following descriptor system described by

Ei =Ax + Bu,

y==cx+Du,

where E ,g Bnxn, A E Rnx”, B E Rnx(P+q), C E R(m+r)xn, D E R(m+r)X(p+q). we assume the system (6) is regular, i.e., --SE + A is a

regular pencil.

(J, J’)-LOSSLESS FACTORIZATION 1293

The system (6) is said to be controllable (respectively, observable) if rank[ SE - A B ] = n (respectively, ranHsET - AT CT] = n) holds for any s E C u {m}. The controllability at infinity can be checked by the following way [23]: find a nonsingular matrix M satisfying

(sE -A)M = [sE, -A, AZ],

where E, is of full column rank; then the system is controllable at infinity if and only if rank[Er A, B] = n. The system (6) is minimal if it is controllable and observable.

Similarly, the system is said to be stabilizable (respectively, detectable) if rank[ SE - A B] = n (respectively, rank[ sET - AT CT ] = n> holds for any s E c+u 0,.

The identity

[+=j-j = [q], IMlfO, lNl#O,

(7)

is termed a restricted equivalent transformation.

2.2. Q,]‘JLO 1 ss ess Factorization and J-Lossless Conjugation

Here we give a descriptor-form characterization of (1, I’)-lossless matrices which is obtained by a slight modification matrices [ 151.

LEMMA 1. O(s) is <J, J’>-lossless ifits reakzation

O(s) = [+q$]

satisfies

ATH + HTA + C’JC = 0,

BTH + D’JC = 0,

ETH= HTE > 0

for a matrix H, and D is (J, ] ‘)-unitary.

_ ”

of that of ]-lossless

(8)

(9)

(10)

(11)


Proof. From the assumption, direct calculation yields

and

J’ - @*JO = 2Re[s] BT(iET - AT))lETH(sE - A)-‘B > 0,

Re[s] > 0.

Thus O(s) is (J, Jr)-lossless. n

Now we give the definition of the (J, I’)-lossless factorization.

DEFINITION 1. G(s) E R(s)(mfr)x(p+9) is said to have a (J, I’)-ZossEess factorization if it is represented as a product

G(s) = O(s)II(s), (12)

where O(s) is (1, J’)-lossless, and II(s) = R(s)(P+~)~(P+~) has neither zeros nor poles in C,.

In the case where G(w) exists and is injective, and G(s) has no poles on the jw-axis, (12) implies that both II(s) and II’(s) are in RH”. Thus, the above definition coincides with the usual (J, Jr)-lossless factorization in [18]. Furthermore, the above definition also allows us to treat the case where G(s) has jo-axis poles. If O(s) and II(s) satisfy (121, so do O(s)D, and D;‘II(s>, where 0, is any constant J’-unitary matrix. Therefore, the (J, J’)-lossless factorization is unique only up to a constant II-unitary matrix.

It is known that J-lossless conjugation is a powerful tool for calculating the J-lossless factorization of a rational function. Here we give the definition of J-lossless conjugation from [20].

DEFINITION 2. A J-lossless matrix O(s) is said to be a stabilizing (antistabilizing ) J-l oss ess 1 conjugator of a rational matrix G(s) if and only if

(i> G(s)@(s) is stable (antistable), (ii> deg O(s) is equal to th e number of unstable (stable) poles of G(s).


Let

G(s) = [w] (13)

be a descriptor-form representation of G(s). We first introduce the stabilizing solution of a generalized Riccati equation (GRE) from [15]. Consider the following set of conditions:

ATXE + ETXA - ETXBRBTXE = 0, (14

IA - BRBTXE - sE/ # 0 Vs E c,, (15)

Ker{ XE} I V{ -sE + A; a,}, (16)

where R is a symmetric matrix. If there exists a symmetric solution X E RnX n

satisfying (14)-(161, then we write

X E GRic{ -SE + A, B, R}.

Note that the above notation is slightly different from the one in [15], where (16) was written as

Ker{ XE} 3 V( -SE + A; C-U Ln,). (17)

In fact, (14) and (15) imply that Ker(XE} 1 Y( -SE + A; C_}. Therefore, (17) can be reduced to (16). In [14], it is pointed out that X is not unique if E is singular, but XE is uniquely determined, and a computation algorithm for the GRE was given.

The following lemmas from [I51 will be useful in performing (J, J’)- lossless factorization later.

LEMMA 2 [15]. Let G(s) be a rational matrix in R(s)(~+~)~(~+‘) whose

stabilizable and detectable realization is given in (13). Then there exists a

stabilizing ]-lossless conjugator O(s) of G(s) if and only if the following statement holds :

X E GRic{ -SE + A, B,J), ETXE > 0. (18)


In this case, a desired conjugator O(s) and the conjugated system G(s)@(s)

are given respectively by

O(S) = [*]Dc (19)

= [W]Dc, (20)

G(s)@(s) = [w]Dr. (21)

where D, is any constant J-unitary matrix, and G(s)@(s) given in (21) is

stabilizable and detectable.

REMARK. The equivalence of two realizations for O(s) in (19) and (20)

can be verified directly with the aid of (18). Moreover, from the realization

(201, O(S) is J-lossless according to Lemma 1.

LEMMA 3 [15]. Let G(s) be a rational matrix in R(s)(ntr)x(P+q) whose stabilizable and detectable realization is given in (13). Then there exists an

antistabilizing ]-lossless conjugator O(s) of G _ (s)] if and only f the

following statements hold:

Y E GRic{ -sET + AT,CT, -J}, EYET > 0. (22)

In this case, a desired conjugator O(s) and the conjugated system II - (s) :=

G” (s)]@(s) are given respectively by

(23)

(24

II(s) = DC’J -SE + A + EYCT]C B + EYCT]D

C 1 D ’ (25)

where D, is any constant ]-unita y matrix, and n(s) given in (25) is

stabilizable and detectable.

<J, J’hLOSSLESS FACTORIZATION 1297

3. SYSTEM MATRICES AND HAMILTONIAN MATRIX PENCILS

The system matrices of G(s) and G - (s)JG(s) are the following matrices:

P(s) = -sPE + PA, W(s) = -SW, + w,, (26)

where

p := E 0 E [ 1 0 0’ A B

P*:= c *, [ 1 (27)

W(s) is usually called a Hamiltonian matrix pencil. For the case E = I, the eigenstructures of P(s) and W(s) h ave been studied. In [12], it was found that the eigenstructure of P(s) on a, can be determined from that of W(s) on a,. In [3], a similar problem was discussed, and the spectral factorization problem was solved for state-space systems. In [24], the eigenstructures of P(s) and W(s) was explored to study the (J, I’)-lossless factorizations for strictly proper functions via the method of infinite zero compensator proposed in [4]. Throughout this paper, we suppose that P(s) has full normal rank of n + p + q, i.e., P(s) is of full column rank.

In [7], an algorithm to compute Kronecker’s canonical form of a singular pencil was proposed. Since PE in (27) has p + q zero columns, the basis of the eigenspace of P(s) on fiR, can be spanned by a matrix of the form

PO 0

[ 1 *I’ P+4

which fact is expressed by

lm( [: zp:q]] = uIP(sL%~. (2%

Here * denotes a block matrix whose precise form is irrelevant to the following discussions. Suppose that the dimension of the stable eigenspace of W(s) is n,, and the corresponding eigenspace is spanned by the matrix


i.e.,

Im (30)

From (30) we see that there exists a strictly Hurwitz A E R”I’“I satisfying

EP

W(s) ; I[ 1 = ET@ ( -sZ + h). u 0

LEMMA 4. The following statements hold:

(i) STEP = PTET@.

(ii) aTAP + PTA@ + PTCTJCP = UTD?jDU. (iii) STEP, = 0.

(iv) Zf [P P,] is nonsingular, ET[@ O][ P P,]-l is symmetric.

(3 1)

Proof. Let A := STEP - PTET@, V := [QT -PT UT]. Multiplyingthe identity (31) at s = 0 by V from the left, we have

QTAP + PTA@ + PTCTJCP - UTDTJDU = AA, (32)

which implies that AA is symmetric. From AT = -A, it follows that

AA + ATA = 0.

Since A is strictly Hurwitz, we have A = 0. This completes the proof of (i). Moreover, (ii) follows immediately from (32).

Rearranging the identity (31) at s = 0, we have

1 E 0 0

= -AT[@T, -PT UT] 0 ET 0 , [ 1 0 0 0

i.e..

VW, = -AT?WE. (33)

(J, I’)-LOSSLESS FACTORIZATION 1299

Decompose P, = [Poj Porn] in (29) so that the eigenspaces of P(s) on R and {m} are spanned by

[ ‘~1 and [‘r If,tq].

respectively. Define

‘0j q := 0

*

P 000 0

T,:= [ 0 0 1 ,

* 1 P+4

where Tj and T, have 2n + p + q rows. Then, from (4), there exists a matrix hi with all its eigenvalues on 1R satisfying

which implies that

WA?; = W,T,A,. (34

Multiplying (34) by V from the left and multiplying (33) by TJ from the right, we have

VWE?jAj + A’VW,?;. = 0.

Since the eigenvalues of A, and -A are different, WETi = 0. It follows

that

(D?‘EP,,j = 0.

Similarly, from (S), there exists a nilpotent matrix N such that

P 0m 0

[ 1 * z . P+q

1300 XIN XIN AND HIDENORI KIMUFtA

It follows that

WATmN = WETm. (35)

By using (33) and (35) repeatedly, we get

VW,T, = VW,T,N = -AT?WET,N = ... = ( -AT)k~E~m~k,

where k is an arbitrary integer. Since N is a nilpotent matrix, N k = 0 will hold if k is big enough. Thus, VW,T, = 0, which implies that

Thus

aTEP, = QTE[ Poj PO,] = 0,

i.e., (iii) holds. Finally, by using (i> and (iii), direct verification yields that

ET[Q, O][P PO]-’

is symmetric, which proves (iv). W

The above method of proof is an extension of the one used in [4] for the case E = I. Suppose that P, has n, columns; then n, + n1 = n if P(s) is square. Indeed, it follows from (30) and (31) that

Im

-sE+A B 0

c D 0 =U 0 0 _&r _ AT -CT 1 ,C_ I . (36)

0 0 -BT -DT

From the regularity of P(s), we know the above statement holds. However, it is not true for general nonsquare matrices. In the next section, we shall prove that no + nl = n holds if G(s) has a (1, Jr)-lossless factorization.


G(s) in (13) is said to be stable if det( -SE + A) # 0 for s E C +. This definition implies that II(s) in (12) and K’(s) are stable. When G(s) is square, we have the following lemma.

LEMMA 5. Suppose a stabilizable and detectable realization (13) of square G(s) is stable. Then the following statements are equivalent:

(i) G(s) has a J-lossless factorization. (ii) There exist matrices X = XT E R” ’ n and Z E R(” + r)x n satisfying

[‘: :T] E Gtio( [ -‘%+ A ;]> [ _e+,],J}. ETXE > 0. (37)

(iii) [P PO] is nonsingular, and ET[@ O][P P,]-’ > 0, where Q,, P, and

P, are defined in (29) and (30).

If one of the above statements holds, then O(s) and II(s) are given by

O(s) = [ -s;+A ; ) ;I]. (38)

H(+=-y] (39)

where

L := [CP + DU O][P P,]-‘, (40)

c^ := C - L=[ -DU CP,][P P,]-‘. (41)

Proof. (i) w (ii): From Lemma 3 in [15] (which is immediate from Lemma 2) G(s) can be factorized as G(s) = @(s)II(s), where O(s) E RH” is J-lossless, II(s) has no zeros in C,, and there is no unstable pole-zero cancellation between O(s) and II(s) if and only if condition (ii) holds. Hence, the stability of G(s) implies that of II(s). Therefore, G(s) has a J-lossless factorization. This completes the proof of the equivalence of conditions (i) and (ii).

(ii) e (iii): From (36) and Theorem 3.1 in [14], the solvability condition for the GRE in (37) is the nonsingularity of [P P,], and when the GRE is solvable, XE = [a O][ P PO]-l. Thus, (ii) and (iii) are equivalent. n


From the definition of the GRE in (14)-(161, P(s) must be square. Therefore, the GRE expressed in (ii) of Lemma 5 is no longer appropriate for treating the nonsquare case, while condition (iii) has no such problem. Therefore, condition (iii), which is confined to generalized eigenvalue problems, is used in the following discussions.

It can be shown in the following lemma that the (J, J’)-lossless factorization for general unstable G(s) can be reduced to that for a certain stable system after the unstable poles have been extracted from G(s).

LEMMA 6 [20]. A (J, J’>-2 oss ess a I f ct orization of G(s) exists if and only if an antistabilizing J-lossless conjugator O+(s) of G - (s>J exists such that

JO; (s)JG(s) has a <J, J’>-Z oss ess a 1 f ct orization O_(s>II(s>. In that case, O(s) = O+(s)@_(s) and II(s) give a <J, J’)-lossless factorization of G(s).

Thus, we shall discuss the (1, J’)-lossless factorization for a stable G(s) in the following section.

4. (J, Jr)-LOSSLESS FACTORIZATION FOR STABLE G(s)

Here we first state the following lemma.

LEMMA 7. Suppose that stable G(s) E R(s)(~+“)~(P+~) with its stabilizable and detectable realization given in (13) has a (J, ]I)-lossless factorka- tion. Then [P P,] is nonsingular, and

ET[Q, O][P P,]-’ > 0,

where P,, a’, and P are defined in (29) and (30).

Proof. Assume that G(s) has a (J, J’klossless factorization G(s) =

O(s)II(s>, where

O(s) = [s], “(s) = [*I, (42)

are a minimal realization for O(s) and a stabilizable and detectable realization for II(s), respectively. Since O(s) is a (1, J’)-lossless matrix, there exists

(1, J’bLOSSLESS FACTORIZATION 1303

an X, > 0 such that

ATX, + X,A, + CT& = 0, (43)

BFX, + D;Jq = 0 (44)

hold [18]. Since G(s) is stable, A, is strictly Hurwitz. We first suppose that m > p, q = r. From [17], we know that there exists

O’(s) such that

O(s) := [O’(s) O(s)] = [fi]

is J-lossless, where D, ’ is the J-orthogonal complement of D,, i.e., [ 0; Dll is J-unitary, and Bi := -X;‘C$D;.

Let e(s) := [O’(s) G(s)]. Then it is obvious that

gives a J-lossless factorization of E(s). Furthermore, G(s) and c(s) have the following descriptor-form realizations:

Now, we will examine the eigenstructures of c(s) in (46) and G(s) in (45). Let

1304 MN XIN AND HIDENORI KIMURA

From

--sE,+Ao B,

ccl %I I

-sZ + A, + X,‘C;JC, 0 0 0

0 -sE,+A, 0 B,

= : D;TJC, 0 Lp 0

I

(48)

J’DTJC, c2 0 D,

and the fact that A, + XL1 CTJCl = -X, ‘ATX, is antistable due to (431, we

have

PO := p” [ I , OTT (49)

where

Let

-SE, + A, 0

w,(s) := - WC0 -SE; - A; (51)

WC0 ET 0

From (43) and (44), we get

I -sI + A, 0 B,Cz 0 B,*, B;

0 -sI -AT 0 0 0 0

0 0 -SE, A, 0 B2 0

0 -CTBT 0 2 1 - C,TJ’C2 -SE; -A$ -C;J’D,

0 DTBT 2 1 *;.l’C, B,T D;T]‘Dz 0

0 BIT 1 0 0 0 L-,:

(52)

<J, J’hLOSSLESS FACTORIZATION 1305

where

0 0 0 0 0 0 I 0 0 0 I 0 0 0 0

T := ooozoo’ oooooz 0000I0

This implies that

(I 0’ 0 0

0 p, Imc 0 an 1 IJ

) = v{T-'W,T,C_},

0 u?r 0 0

(53)

where

-SE, + A, 0

- C,TJ’C, -SE; - A;

WC, *,T

Since II(s) given in (42) has no zeros in C,, n - (s) has no zeros in C-. It

follows that

det -SE; - A; -Bz'

Cz’ E 1 + 0, s E c-, which implies from (36) that

Thus.

@, = 0, C,P,, + D,U,, = 0.

(55)

1306

satisfy

XIN XIN AND HIDENORI KIMURA

P Im @ ([ II = “{Wa(S>, C}.

U

Similarly, we have

(57)

(58)

-SE, + A, 0 BO

- GOTJGO -SE; - A; -C,?jQ, ,C_ ,

WC0 B,T D,‘Po I I

C-59)

where

u := [o u,]. (60)

Since E(s) has a J-lossless factorization, we know that [ P PO I is nonsingular from Lemma 5. From (56) we have

E,T[@ O][P PO]-l = ; ; > 0. [ 1

If the lemma is true for a particular stabilizable and detectable realization of G(s), e.g., the one in (45) it is easy to see that the statement is also true for any stabilizable and detectable realization.

Finally, if q < r, then there exists O”(s) such that [O’(s) O(s) O(s>“l is J-lossless. A similar analysis can be carried out, and it is omitted here. n

Now, we calculate the (1, J’)-1 oss ess 1 factorization for a stable G(s).

LEMMA 8. A stable G(s) with its stabilizable and detectable realization

in (13) has a (J, I’)-lossless factorization if the following statements hold:

(i) [P PO] is nonsingular, and ET[@ Ol[ P PO]-’ z 0, where PO, a!, and

P are dejned in (29) and (30).

(J, J’)-LOSSLESS FACTORIZATION 1307 .

(ii) There exist D, E R(p+q)x(p+q), F E R(pi-q)xn, and a (],J’>-lossless

matrix D, such that

[c^ D] = D,[F D,], (61)

,. where C is &fined in (41). In this case,

@(s)=[ -‘!” !i -{+q] (62)

= [-‘B’* !m 1 -i+q], (63)

l-qs) = [q+]> (64

where L is defined in (40).

Proof. Assume that (i) and (ii) hold, and let H := [(a 0][ P P,]-l. From Lemma 4,

ATH + HTA + C’]C = cTJc” = FT]‘F,

BTH + D’JC = D’Jc^ = D,‘J’F,

ETH = HTE > 0.

Direct calculation yields

G” (s)]G(s) = [ --’ -SE;; AT 1 ;$D].

(65)

(66)

(67)

(68)

1308 KIN KIN AND HIDENORI KIMURA

Taking

in the transformation (7) and with (65)-(67) yields

G” (s)JG(s)

(69)

which implies that O(s) = G(s)ll-‘(s) is (J, J’)-unitary with the realization (62). The equivalence of the two realizations for O(s) in (62) and (63) follows directly from (41). Denote (61) as

@(s) = I-““‘” :* 1 -;+J [+!j__+ (76)

1 L OI RI and define

H 0

H ‘= DC’JL 0 -[ 1

Thus, kTG = l? Tz = diag( ETH, 0) 2 0 from ETH > 0. According to Lemma 1, simple calculation yields that O(s) in (70) is (J, ]‘)-lossless. Since G(s) is stable, all finite poles of IT(s) are obviously stable. From G(s) = @(s)I’I(s) and the fact that O(s) has neither zeros nor poles in R,, the dimensi:n of the eigenspace on fi, of n(s) is the same as that of G(s). From CP + DU = 0 and (61), it follows that FP + D,U = 0 holds, which

(1, J’)-LOSSLESS FACTORIZATION

implies together with (31) that

-sE+A B F 077 I[

1309

P u 1 = "0' (-sZ+A), [ 1 (71)

i.e., all the finite zeros of II(s) are stable, which completes the proof. W

Due to Lemma 7, condition (i) of Lemma 8 is also a necessary condition for G(s) to have a (J, Jr>-1 oss ess 1 factorization. Condition (ii) is the (J, I’>- lossless factorization of a given constant matrix; however, it is not a necessary condition. Indeed,

G(s)=[ -s;+A iX in] (72)

is also a stabilizable and detectable realization of G(s) in (13), where C, is an arbitrary matrix with appropriate dimension. If condition (ii) held for the realization (72), it would follow that C, = D,C,. That is obviously not true, since C, is arbitrary. Therefore, we must consider a more restricted class of plants in order to make condition (ii) necessary.

Actually, the realization of G(s) in (72) is only the realization in (13) with nondynamical variables added. To exclude the trivial augmentation or defla- tion of nondynamic variables, we consider a special kind of realization of

G(s).

DEFINITION 3 [23]. G(s) is said to be in the standard firm if it is impossible to find nonsingular matrices M and N for which

M( -SE + A)N = -&+A^ 0 0 1 z *

Now we state one of the main results of the paper.

THEOREM 1. Zf a stable G(s) has a stabilizable and oktectable realization in (13) which is in standard form, then G(s) has a (J, I’)-lossless factorization if and only if conditions (i) and (ii) in Lemma 8 hold.

Proof. From Lemmas 7 and 8, we only have to prove that condition (ii) in Lemma 8 is a necessary condition. Suppose that -SE, + A, in (42) is in


the standard form. Therefore, G(s) in (45) is in standard form. From (45) D, = DID,, and

c^, = [ -D,U C,P,][P PO]-’

= D,[O -D,K Wo,l[P W1,

where Pa, P, and U are defined in (49) (56) and (60) respectively. Since D, is (1, Jr)-lossless from (42), then condition (ii) holds for G(s) in (45). Since the realization of G(s) is in standard form, the D-matrix of G(s) is invariant under the different realization, and with the aid of Lemmas Al and A2 in the Appendix, condition (ii) holds for a minimal realization in the standard form of G(s). Moreover, since any two minimal realizations in the standard form of G(s) are restricted equivalent [23], i.e., there exist nonsingular matrices M and N such that two realizations of G(s) satisfy

-SE, + A, B,

C, Di] = [: ‘:I[ _%+A :I[: ;I7 (74)

and it is obvious that the validity of condition (ii) is invariant under a restricted equivalent transformation. Therefore, any stabilizable and detectable realization in the standard form satisfies condition (ii) if stable G(s) has a (J, J’)-lossless factorization. W

5. (J, J’)-LOSSLESS FACTORIZATION FOR GENERAL UNSTABLE G(s)

Suppose that G(s) with its stabilizable and detectable realization has a (1, I’)-lossless factorization G(s) = @(s)II(s). According to Lemma 6, G _ (s)J has an antistable J-lossless conjugator O+(s). Due to Lemma 3, 0 + (s) is calculated to be

@+(s) = [ -;+A 1 -7”*q, (75)

where

Y E GRic{ -sET + AT, CT, -J}, EYE* > 0. (76)


Also,

G_(s) := I@; (s)JG(s) = -‘.E + “c’ EYCTIC B + ‘F’JD . I

The realization of G(s) in (77) is stabilizable and detectable according to Lemma 3, and G_(s) is stable according to Lemma 6. Thus, we shall derive a (J, J’)-lossless factorization of G_(s) by using Lemmas 7 and 8.

For simplicity of notation, let H, := YET. Since

= -SE + A + H;CTJC B + H,TCTJD

c I D ’ (78)

it follows from (27) that

=V -SE + A + H;CT]C B + H,TCTJD

c D 1 I >fi, * (7%

Let

W”(S)

-SE + A + H;CTJC 0 B + H;CT]D

:= - C’]C -sET - AT - CTJCH, -CTJD .

D’JC BT + DTJCH, D’JD 1

1312

By direct calculation, we note that

[; f JW&$

where W(s) is defined in (26). Thus

KIN KIN AND HIDENORI KIMURA

-H, 0

z 0

I

= W(s), (80) 0 I

[;I= [I f” Bl[d] = [‘-{“I (81)

satisfies pw Im @,

i[ II = ~{WJs),c-}> (821

ZJ!,

where P, @, and U are defined in (28). From (81), we know that the generalized eigenvalue problem (82) d p e en d s on the solution of the GRE of

(76). We will decouple the generalized eigenvalue problem (82) from (76) as follows, under the assumption of E2 = E.

From Lemmas 7 and 4, [P, P,,] is nonsingular, and E*H, is symmetric and nonnegative, where

H, := [@ O][P - H,@ P,,]?

Therefore,

I + EYE*E*H > 0 W .

By using the identities det(Z + AB) = de6 Z + BA) and det A = det A*, together with E2 = E, we get

det( Z + EYE*E*H,) = det( Z + YE*H,E) = det( Z + YHZE)

= det( Z + E*H,Y) = det( I + H,YE*)

= det( Z + H, H,) # 0, (83)

(J, I’)-LOSSLESS FACTORIZATION

which implies that

[P PO] = (z + H&J[P - H,Q, P,]

is nonsingular. Define

1313

(84)

from (84), we get

H, := (Z + H,H,)-lH,; (85)

H, = [@ O][P P,]-‘. (86)

From Lemma 4, ETH, is symmetric. Multiplying (85) by ET from the left, we have

ETH, = ET(Z + HwYETET)-1 H, = (I + ETH,YET)-lETHw

= (Z + H;EEYET)-lETHw = (Z + ETHwEYET)-lETHw z 0.

(87)

Therefore, from

ETHw = (Z - ETH,EYET)-lETH, > 0,

it follows that

p( ETH, EYET) < 1. (88)

Now, we analye condition (ii) in Lemma 8. According to (41), let

& := c - [C(P -H,@) + DU O][P - H,@ P,]-’

= c^( Z - H, H,)-‘, (89)

where 6 is defined in (41). If there exist D, E R(P+9)X(P+9), F E R(P+9)X”, and (J, I’)-lossless matrix D, such that

[6 D] = D,[F D,] (90)


holds, then there exists F, E R(p+qjXn such that

holds, where

F, := F(Z - H,H,)-l. (92)

Therefore, according to Lemmas 8 and 6, we have

rI(s) =

I

--SE + A + H;CTJC B + H;CT]D Fw 1 Q? ’ (93) --SE + A + H;CTJC B + H,TCT]D 0

O_(s) = FW D7r -Zp+q 1 (94)

C D 0 I

and

In summary, we have the following lemma.

LEMMA 9. Suppose G(s) with its realization in (13) is stabilizable and

detectable, and E2 = E holds. Then G(s) has a <J, J’)-lossless factorization if

the following statements hold:

(i) [ P pO] is nonsingular, and ET[@ OIL P PO I- ’ > 0, where P, and P

are defined in (29) and (SO), respectively. (ii) Y E GRic{ - SET + AT, CT, -J}, EYET > 0.

(iii) p(ET[@ O][P Z’,lmlEYET> < 1. (iv) There &St 0, E R(P+q)x(P+q), F E R(p+qjxn, and a (J, I’>-

lossless matrix D, satisfying

[c^ D] = D,[F D,],

where c^ is defined in (41).

(96)

(],I’)-LOSSLESS FACTORIZATION 1315

In this case, O(s) and II(s) are given by (95) and (93).

Due to Lemma 7, the following lemma is immediate from Lemma 9.

LEMMA 10. Conditions (i>-(iii) in Lemma 9 are also necessary condi-

tions for G(s) to have a (J, ]‘I-lossless factorization.

REMARK. For the J-lossless factorization, i.e., J = J’, condition (iv) in Lemma 9 is redundant. Indeed, D, := I,,,, F := c^, and D, := D satisfy the condition. Thus, conditions (i>-(“‘> m are necessary and sufficient for a square matrix G(s) to have a J-lossless factorization. This is the exact result which has been stated by Theorem 5 in [15].

Now, we state a main result of this paper, which follows directly from Theorem 1 and Lemma 9.

THEOREM 2. Suppose G(s) with its realization in (13) is stabilizable and

detectable and is in standard form, and suppose E2 = E. Then G(s) has a

oss ess a 1 f ct otization if and only if conditions (i)-(iv) in Lemma 9

REMARK. Given a descriptor-form realization of G(s), we can get its standard form by deflating the modes of nondynamic variables after using the restricted transformation in (7). Note that E = Z is a special case of the standard form. Thus, Theorem 2 gives a necessary and sufficient condition for the (J, J’>-1 oss ess 1 factorization of a strictly proper function with jw-axis zeros, which includes the result of [24]. Here, the approach in [24] using the method of infinite zero compensation derived in [4] is avoided.

6. CONCLUSION

The (J, I’>-1 oss ess 1 factorization for descriptor systems based on the theory of conjugation has been studied in this paper. Necessary and sufficient conditions have been proposed via the generalized eigenvalue problems associated with the system matrix and Hamiltonian matrix pencil of the descriptor system. For the (J, 1’1-1 oss ess 1 factorization of a special descriptor system arising from a nonstandard H” control problem, it is interesting to find that condition (iv) in Lemma 9 is satisfied under conditions (i)-(iii). Therefore, Lemma 9 is a necessary and sufficient condition in such a case. The application of the (1, J’>-1 oss ess 1 factorization to 4-block H” control problems with jw-axis zeros, which is an extension of the approach of [20] to the nonstandard case, will be reported later.


APPENDIX

LEMMA Al. Condition (ii) in Lemma 8 holds for the realization

[ -‘;; *’ -+ 1 ;] (Al)

if and only if it holds for the realization

where A, is stable and (E,, A,, B2) is controllable.

Proof. Since

0 0 Im P, 0 =p

i[ I) 0 I

where

-sZ + A, 0 0

4 -sE,+A, B,

Cl c2 D II ,

Im([: :I) =v{[ -sE12+A2 _2]),

then

[c, c21 i. = GPO, [ 1

which is independent of C, associated with the uncontrollable mode -sI + A,. This completes the proof. n

(J,J’)-LOSSLESS FACTORIZATION

Dually, we have

LEMMA A2. Condition (ii) in Lemma 8 holds for the realization

[ -“fA1 -s;:A2 ’ ;]

if and only if it holds for the realization

[ -sE;:A, I;],

where A, is stable and (E,, A,, C,) is observable.

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Received 28 Februa y 1993; final menuscript accepted 2 October 1993

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